Fairness Guarantees in Allocation Problems

Fair division problems have been vastly studied in the past 60 years. This line of research was initiated by the work of Steinhaus in 1948 in which the author introduced the cake cutting problem as follows: given a heterogeneous cake and a set of agents with different valuation functions, the goal is to find a fair allocation of the cake to the agents. In order to study this problem, several notions of fairness are proposed, the most famous of which are proportionality and envy-freeness, introduced by Steinhaus in 1948 and Foley in 1967. The fair allocation problems have been studied in both divisible and indivisible settings. For the divisible setting, we explore the "Chore Division Problem". The chore division problem is the problem of fairly dividing an object deemed undesirable among a number of agents. The object is possibly heterogeneous, and hence agents may have different valuations for different parts of the object. Chore division is the dual problem of the celebrated cake cutting problem. We give the first discrete and bounded envy-free chore division protocol for any number of agents. For the indivisible setting, we use the maximin share paradigm introduced by Budish as a measure of fairness. We improve previous results on this measure of fairness in the additive setting and generalize our results for submodular, fractionally subadditive, as well as subadditive settings. We also model the maxmin share fairness paradigm for indivisible goods with different entitlements. For the indivisible setting, we also consider the most studied notion of fairness, envy-freeness. It is known that envy-freeness cannot be always guaranteed in the allocation of indivisible items. We suggest envy-freeness up to a random item (EFR) property which is a relaxation of envy-freeness up to any item (EFX) and give an approximation guarantee. For this notion, we provide a polynomial-time 0.72-approximation allocation algorithm

A Discrete and Bounded Envy-free Cake Cutting Protocol for Four Agents

We consider the well-studied cake cutting problem in which the goal is to identify a fair allocation based on a minimal number of queries from the agents. The problem has attracted considerable attention within various branches of computer science, mathematics, and economics. Although, the elegant Selfridge-Conway envy-free protocol for three agents has been known since 1960, it has been a major open problem for the last fifty years to obtain a bounded envy-free protocol for more than three agents. We propose a discrete and bounded envy-free protocol for four agents

An Algorithmic Framework for Strategic Fair Division

We study the paradigmatic fair division problem of allocating a divisible good among agents with heterogeneous preferences, commonly known as cake cutting. Classical cake cutting protocols are susceptible to manipulation. Do their strategic outcomes still guarantee fairness? To address this question we adopt a novel algorithmic approach, by designing a concrete computational framework for fair division---the class of Generalized Cut and Choose (GCC) protocols}---and reasoning about the game-theoretic properties of algorithms that operate in this model. The class of GCC protocols includes the most important discrete cake cutting protocols, and turns out to be compatible with the study of fair division among strategic agents. In particular, GCC protocols are guaranteed to have approximate subgame perfect Nash equilibria, or even exact equilibria if the protocol's tie-breaking rule is flexible. We further observe that the (approximate) equilibria of proportional GCC protocols---which guarantee each of the $n$ agents a $1/n$-fraction of the cake---must be (approximately) proportional. Finally, we design a protocol in this framework with the property that its Nash equilibrium allocations coincide with the set of (contiguous) envy-free allocations

On the Complexity of Chore Division

We study the proportional chore division problem where a protocol wants to divide an undesirable object, called chore, among $n$ different players. The goal is to find an allocation such that the cost of the chore assigned to each player be at most $1/n$ of the total cost. This problem is the dual variant of the cake cutting problem in which we want to allocate a desirable object. Edmonds and Pruhs showed that any protocol for the proportional cake cutting must use at least $\Omega(n \log n)$ queries in the worst case, however, finding a lower bound for the proportional chore division remained an interesting open problem. We show that chore division and cake cutting problems are closely related to each other and provide an $\Omega(n \log n)$ lower bound for chore division

Efficient Algorithms for Envy-Free Stick Division With Fewest Cuts

Given a set of n sticks of various (not necessarily different) lengths, what is the largest length so that we can cut k equally long pieces of this length from the given set of sticks? We analyze the structure of this problem and show that it essentially reduces to a single call of a selection algorithm; we thus obtain an optimal linear-time algorithm. This algorithm also solves the related envy-free stick-division problem, which Segal-Halevi, Hassidim, and Aumann (AAMAS, 2015) recently used as their central primitive operation for the first discrete and bounded envy-free cake cutting protocol with a proportionality guarantee when pieces can be put to waste.Comment: v3 adds more context about the proble